Forecasting Time Series with VARMA Recursions on Graphs

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This work provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., timevarying graph signals whose first and second order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with VAR and VARMA models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain: (i) the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate ARMA models and (ii) an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin

The Log-GARCH Model via ARMA Representations

The log-GARCH model provides a flexible framework for the modelling of economic uncertainty, financial volatility and other positively valued variables. Its exponential specification ensures fitted volatilities are positive, allows for flexible dynamics, simplifies inference when parameters are equal to zero under the null, and the log-transform makes the model robust to jumps or outliers. An additional advantage is that the model admits ARMA-like representations. This means log-GARCH models can readily be estimated by means of widely available software, and enables a vast range of well-known time-series results and methods. This chapter provides an overview of the log-GARCH model and its ARMA representation(s), and of how estimation can be implemented in practice. After the introduction, we delineate the univariate log-GARCH model with volatility asymmetry ("leverage"), and show how its (nonlinear) ARMA representation is obtained. Next, stationary covariates ("X") are added, before a first-order specification with asymmetry is illustrated empirically. Then we turn our attention to multivariate log-GARCH-X models. We start by presenting the multivariate specification in its general form, but quickly turn our focus to specifications that can be estimated equation-by-equation - even in the presence of Dynamic Conditional Correlations (DCCs) of unknown form. Next, a multivariate non-stationary log-GARCH-X model is formulated, in which the X-covariates can be both stationary and/or nonstationary. A common critique directed towards the log-GARCH model is that its ARCH terms may not exist in the presence of inliers. An own Section is devoted to how this can be handled in practice. Next, the generalisation of log-GARCH models to logarithmic Multiplicative Error Models (MEMs) is made explicit. Finally, the chapter concludes

The Log-GARCH Model via ARMA Representations

The log-GARCH model provides a flexible framework for the modelling of economic uncertainty, financial volatility and other positively valued variables. Its exponential specification ensures fitted volatilities are positive, allows for flexible dynamics, simplifies inference when parameters are equal to zero under the null, and the log-transform makes the model robust to jumps or outliers. An additional advantage is that the model admits ARMA-like representations. This means log-GARCH models can readily be estimated by means of widely available software, and enables a vast range of well-known time-series results and methods. This chapter provides an overview of the log-GARCH model and its ARMA representation(s), and of how estimation can be implemented in practice. After the introduction, we delineate the univariate log-GARCH model with volatility asymmetry ("leverage"), and show how its (nonlinear) ARMA representation is obtained. Next, stationary covariates ("X") are added, before a first-order specification with asymmetry is illustrated empirically. Then we turn our attention to multivariate log-GARCH-X models. We start by presenting the multivariate specification in its general form, but quickly turn our focus to specifications that can be estimated equation-by-equation - even in the presence of Dynamic Conditional Correlations (DCCs) of unknown form. Next, a multivariate non-stationary log-GARCH-X model is formulated, in which the X-covariates can be both stationary and/or nonstationary. A common critique directed towards the log-GARCH model is that its ARCH terms may not exist in the presence of inliers. An own Section is devoted to how this can be handled in practice. Next, the generalisation of log-GARCH models to logarithmic Multiplicative Error Models (MEMs) is made explicit. Finally, the chapter concludes

On the identification and parametric modelling of offshore dynamic systems

This thesis describes an investigation into the analysis methods arising from identification aspects of the theory of dynamic systems with application to full-scale offshore monitoring and marine environmental data including target spectra. Based on the input and output of the dynamic system, the System Identification (SI) techniques are used first to identify the model type and then to estimate the model parameters. This work also gives an understanding of how to obtain a meaningful matching between the target (power spectra or time series data sets) and SI models with minimal loss of information. The SI techniques, namely. Autoregressive (AR), Moving Average (MA) and Autoregressive Moving Average (ARMA) algorithms are formulated in the frequency domain and also in the time domain. The above models can only be economically applicable provided the model order is low in the sense that it is computationally efficient and the lower order model can most appropriately represent the offshore time series records or the target spectra. For this purpose, the orders of the above SI models are optimally selected by Least Squares Error, Akaike Information Criterion and Minimum Description Length methods. A novel model order reduction technique is established to obtain the reduced order ARMA model. At first estimations of higher order AR coefficients are determined using modified Yule-Walker equations and then the first and second order real modes and their energies are determined. Considering only the higher energy modes, the AR part of the reduced order ARMA model is obtained. The MA part of the reduced order ARMA model is determined based on partial fraction and recursive methods. This model order reduction technique can remove the spurious noise modes which are present in the time series data. Therefore, firstly using an initial optimal AR model and then a model order reduction technique, the time series data or target spectrum can be reduced to a few parameters which are the coefficients of the reduced order ARMA model. The above univariate SI models and model order reduction techniques are successfully applied for marine environmental and structural monitoring data, including ocean waves, semi-submersible heave motions, monohull crane vessel motions and theoretical (Pierson- Moskowitz and JONSWAP) spectra. Univariate SI models are developed based on the assumption that the offshore dynamic systems are stationary random processes. For nonstationary processes, such as, measurements of combined sea waves and swells, or coupled responses of offshore structures with short period and long period motions, the time series are modelled by the Autoregressive Integrated Moving Average algorithms. The multivariate autoregressive (MAR) algorithm is developed to reduce the time series wave data sets into MAR model parameters. The MAR algorithms are described by feedback weighting coefficients matrices and the driving noise vector. These are obtained based on the estimation of the partial correlation of the time series data sets. Here the appropriate model order is selected based on auto and cross correlations and multivariate Akaike information criterion methods. These algorithms are applied to estimate MAR power spectral density spectra and then phase and coherence spectra of two time series wave data sets collected at a North Sea location. The estimation of MAR power spectral densities are compared with spectral estimates computed from a two variable fast Fourier transform, which show good agreement

Approximate State Space Modelling of Unobserved Fractional Components

We propose convenient inferential methods for potentially nonstationary multivariate unobserved components models with fractional integration and cointegration. Based on finite-order ARMA approximations in the state space representation, maximum likelihood estimation can make use of the EM algorithm and related techniques. The approximation outperforms the frequently used autoregressive or moving average truncation, both in terms of computational costs and with respect to approximation quality. Monte Carlo simulations reveal good estimation properties of the proposed methods for processes of different complexity and dimension

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics

Outlier detection in multivariate time series via projection pursuit

This article uses Projection Pursuit methods to develop a procedure for detecting outliers in a multivariate time series. We show that testing for outliers in some projection directions could be more powerful than testing the multivariate series directly. The optimal directions for detecting outliers are found by numerical optimization of the kurtosis coefficient of the projected series. We propose an iterative procedure to detect and handle multiple outliers based on univariate search in these optimal directions. In contrast with the existing methods, the proposed procedure can identify outliers without pre-specifying a vector ARMA model for the data. The good performance of the proposed method is verified in a Monte Carlo study and in a real data analysis